On relationship ofMC- flows and MAP- flows of events.

The MAP-flow of events with the intensity representing piecewise-constant random process ƒ(t) with two conditions ƒ(t)= ƒ1 or ƒ(t)=ƒ2 (ƒ1> ƒ2) is considered. Process ƒ (t) is unobserveble.Durations of stay of the process ƒ (t) in i condition is a random variable with an exponential distributionfunction. At the moment of the termination of i condition of the process ƒ(t) the followingsituations are possible, each of them occurs instantly: 1) process ƒ(t) passes from i conditionto i and there comes flow of the event in м condition; joint probability of this situation isP(ƒi  ƒi, 1) = P1(ƒi | ƒi), i=1,2; 2) process ƒ(t) passes from i condition to j and there comes flowof the event; joint probability of this situation is P(ƒi  ƒj, 1) = P1(ƒj | ƒi) (i, j = 1, 2; i ≠ j);3) process ƒ(t) passes from i condition to j and the flow of event does not come; joint probabilityof this situation is P (ƒi  ƒj, 0) = P0 (ƒj | ƒi) (i, j=1, 2; i≠j). Thus, P1 (ƒi | ƒi) + P1 (ƒj | ƒi) ++ P0 (ƒj | ƒi) = 1; i, j=1, 2; i≠j.In the definition provided for the MAP-flow nothing is mentioned about such fact as in whatcondition of the process ƒ(t) the flow of the event appears during transformation process ƒ(t) fromthe first (second) condition into the second (in the first). Thereof, depending on what is initial atthe moment of the termination of either this or that condition of the process ƒ(t) - event of theflow or transition from one condition to onother, there are distinguished four kinds of MAP-flowof events of the first order (МАР-1): МАР-1Е, МАР-1Т, МАР-1ЕТ, МАР-1ТЕ (Е - Еvent, T -Transition). The randomized model of the MAP-flow of events of the first order is entered aswell: МАР-1R (R - Random).It is shown that the synchronous MC-flow of events is a special case of МАР-1Е flow.Let's consider two MAP-flows of the first order: МАР(1)-1, МАР(2)-1, where flow МАР(1)-1 isdescribed above, and flow МАР(2)-1 differs from the flow МАР(1)-1 only by the fact that the durationof stay of the process ƒ(t) in i condition is distributed according the law(2)( ) 1 it, 1,2.Fi t = −e−ƒ i= Then, the MAP-flow of the second order is defined as superposition(simple sum) of two MAP-flows of the first order: МАР(1)-1, МАР(2)-1. As each of the specifiedMAP-flows of the first order has four kinds, then MAP-flows of the second order will have 16kinds. We will designate the flow МАР(k)-1Е through МАР1(k); flow МАР(k)-1Т through МАР2(k);flow МАР(k)-1ЕТ though МАР3(k); flow МАР(k)-1ТЕ through МАР4(k); k =1,2. Each kind of theMAP-flow of the second order we will designate by pair MAP-flows of the first order:(MAPi(1) ,MAP(j2)), i,j =1,4.All kinds of MAP-flows of the second order are represented in the form of matrix МАР-2 == ( (1) (2)) 4MAPi ,MAPj 1 . The element of matrix МАР-2 corresponds to the MAP-flow of secondorder МАР(i,j)-2 =(MAPi(1) ,MAP(j2)), i,j =1,4. The randomized model of MAP-flow of events ofthe second order as superposition of randomized models of MAP-flows of the first order is introducedaw well: МАР-2R = ( МАР(1)-1R, МАР(2)-1R ).It is shown that 1) asynchronous MC-flow is a special case of the flow МАР(1,1)-2 or flowМАР(1,2)-2; 2) generalized asynchronous MC-flow is a special case of the flow MC-flow;3) semisynchronous MC-flow is a special case of the flow МАР(1,1)-2 or МАР(1,2)-2 flow;4) generalized semisynchronous MC-flow is a special case of МАР(1,2)-2 flow.

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